Every separable Banach space has a bounded strong norming biorthogonal sequence which is also a Steinitz basis

Terenzi, Paolo

Terenzi, Paolo

Studia Mathematica, Tome 108 (1994), p. 207-222 / Harvested from The Polish Digital Mathematics Library

Résumé

Every separable, infinite-dimensional Banach space X has a biorthogonal sequence $z_{n}, z *_{n}$ , with $s p a n z *_{n}$ norming on X and $∥ z_{n} ∥ + ∥ z *_{n} ∥$ bounded, so that, for every x in X and x* in X*, there exists a permutation π(n) of n so that $x \in \bar{c o n v} f i n i t e s u b s e r i e s o f \sum_{n = 1}^{\infty} z *_{n} (x) z_{n} a n d x *_{n} (x) = \sum_{n = 1}^{\infty} z *_{π (n)} (x) x * (z_{π (n)})$ .

Publié le : 1994-01-01

Zbl 0805.46018

EUDML-ID : urn:eudml:doc:216129

@article{bwmeta1.element.bwnjournal-article-smv111i3p207bwm,
     author = {Paolo Terenzi},
     title = {Every separable Banach space has a bounded strong norming biorthogonal sequence which is also a Steinitz basis},
     journal = {Studia Mathematica},
     volume = {108},
     year = {1994},
     pages = {207-222},
     zbl = {0805.46018},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv111i3p207bwm}
}

Terenzi, Paolo. Every separable Banach space has a bounded strong norming biorthogonal sequence which is also a Steinitz basis. Studia Mathematica, Tome 108 (1994) pp. 207-222. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv111i3p207bwm/

Bibliographie

[00000] [1] S. Banach, Théorie des opérations linéaires, Chelsea, New York, 1932. | Zbl 0005.20901

[00001] [2] W. J. Davis and W. B. Johnson, On the existence of fundamental and total bounded biorthogonal systems in Banach spaces, Studia Math. 45 (1973), 173-179. | Zbl 0256.46026

[00002] [3] W. J. Davis and I. Singer, Boundedly complete M-bases and complemented subspaces in Banach spaces, Trans. Amer. Math. Soc. 175 (1973), 187-194. | Zbl 0256.46027

[00003] [4] A. Dvoretzky, Some results on convex bodies and Banach spaces, in: Proc. Internat. Sympos. on Linear Spaces, Jerusalem Academic Press, 1961, 123-160. | Zbl 0119.31803

[00004] [5] P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309-317. | Zbl 0267.46012

[00005] [6] V. P. Fonf, Operator bases and generalized summation bases, Dokl. Akad. Nauk Ukrain. SSR Ser. A 1986 (11), 16-18 (in Russian). | Zbl 0629.46012

[00006] [7] V. I. Gurarii and M. I. Kadec, On permutations of biorthogonal decompositions, Istituto Lombardo, 1991.

[00007] [8] E. Indurain and P. Terenzi, A characterization of basis sequences in Banach spaces, Rend. Accad. dei XL 18 (1986), 207-212. | Zbl 0618.46014

[00008] [9] M. I. Kadec, Nonlinear operator bases in a Banach space, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 2 (1966), 128-130 (in Russian).

[00009] [10] M. I. Kadec and A. Pełczyński, Basic sequences, biorthogonal systems and norming sets in Banach and Fréchet spaces, Studia Math. 25 (1965), 297-323 (in Russian). | Zbl 0135.34504

[00010] [11] G. W. Mackey, Note on a theorem of Murray, Bull. Amer. Math. Soc. 52 (1946), 322-325. | Zbl 0063.03692

[00011] [12] P. Mankiewicz and N. J. Nielsen, A superreflexive Banach space with a finite dimensional decomposition so that no large subspace has a basis, Odense University Preprints, 1989. | Zbl 0721.46007

[00012] [13] A. Markushevich, Sur les bases (au sens large) dans les espaces linéaires, Dokl. Akad. Nauk SSSR 41 (1943), 227-229. | Zbl 0061.24701

[00013] [14] A. M. Olevskiĭ, Fourier series of continuous functions with respect to bounded orthonormal systems, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 387-432.

[00014] [15] R. I. Ovsepian and A. Pełczyński, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in $L^{2}$ , Studia Math. 54 (1975), 149-159. | Zbl 0317.46019

[00015] [16] A. Pełczyński, All separable Banach space admit for every ε > 0 fundamental total and bounded by 1 + ε biorthogonal sequences, ibid. 55 (1976), 295-304. | Zbl 0336.46023

[00016] [17] A. Plans and A. Reyes, On the geometry of sequences in Banach spaces, Arch. Math. (Basel) 40 (1983), 452-458. | Zbl 0517.46003

[00017] [18] W. H. Ruckle, Representation and series summability of complete biorthogonal sequences, Pacific J. Math. 34 (1970), 511-528. | Zbl 0202.39404

[00018] [19] W. H. Ruckle, On the classification of biorthogonal sequences, Canad. J. Math. 26 (1974), 721-733. | Zbl 0282.46013

[00019] [20] I. Singer, Bases in Banach Spaces II, Springer, 1981. | Zbl 0467.46020

[00020] [21] S. Szarek, A Banach space without a basis which has the bounded approximation property, Acta Math. 159 (1987), 81-98. | Zbl 0637.46013

[00021] [22] P. Terenzi, Representation of the space spanned by a sequence in a Banach space, Arch. Math. (Basel) 43 (1984), 448-459. | Zbl 0581.46010

[00022] [23] P. Terenzi, On the theory of fundamental norming bounded biorthogonal systems in Banach spaces, Trans. Amer. Math. Soc. 299 (1987), 497-511. | Zbl 0621.46013

[00023] [24] P. Terenzi, On the properties of the strong M-bases in Banach spaces, Sem. Mat. Garcia de Galdeano, Zaragoza, 1987.